Lobachevskii Journal of Mathematics

, Volume 34, Issue 2, pp 142–147 | Cite as

Starlike cases of the generalized goodman conjecture

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Abstract

We consider functions f that are meromorphic and univalent in the unit disc \(\mathbb{D}\) with a simple pole at the point p ∈ (0, 1) and normalized by f(0) = f′(0) − 1 = 0. A function g is called subordinated under such a function f, if there exists a function ω holomorphic in \(\mathbb{D}\), ω(\(\mathbb{D}\)) ⊂ Open image in new window , such that g(z) = f((z)), z\(\mathbb{D}\), and we use the abbreviation gf to indicate this relationship between two functions. We conjectured that for gf, the inequalities
$$|a_n (g)| \leqslant \frac{1} {{p^{n - 1} }}\sum\limits_{k = 0}^{n - 1} {p^{2k} } ,n \in \mathbb{N}, $$
are valid. Here f is as above and the expansion
$$g(z)\sum\limits_{n = 1}^\infty {a_n (g)z^n } $$
is valid in some neighbourhod of the origin. In the present article, we prove that this is true for two classes of functions f for which Open image in new window \f(\(\mathbb{D}\)) is starlike.

Keywords and phrases

Starlike meromorphic function subordination 

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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Depart. of Mech. and Math.Kazan Federal UniversityKazanRussia
  2. 2.Institut für Analysis and AlgebraTU BraunschweigBraunschweigGermany

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